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Permutations And Combinations
MODULE - I
Algebra
7
Notes
PERMUTATIONS
AND COMBINATIONS
The other day, I wanted to travel from Bangalore to Allahabad by train. There is no
direct train from Bangalore to Allahabad, but there are trains from Bangalore to Itarsi
and from Itarsi to Allahabad. From the railway timetable I found that there are two trains
from Bangalore to Itarsi and three trains from Itarsi to Allahabad. Now, in how many
ways can I travel from Bangalore to Allahabad?
There are counting problems which come under the branch of Mathematics called
combinatorics.
Suppose you have five jars of spices that you want to arrange on a shelf in your kitchen.
You would like to arrange the jars, say three of them, that you will be using often in a
more accessible position and the remaining two jars in a less accessible position. In how
many ways can you do it?
In another situation suppose you are painting your house. If a particular shade or colour
is not available, you may be able to create it by mixing different colours and shades.
While creating new colours this way, the order of mixing is not important. It is the
combination or choice of colours that determine the new colours; but not the order of
mixing.
To give another similar example, when you go for a journey, you may not take all your
dresses with you. You may have 4 sets of shirts and trousers, but you may take only 2
sets. In such a case you are choosing 2 out of 4 sets and the order of choosing the sets
doesn’t matter. In these examples, we need to find out the number of choices in which it
can be done.
In this lesson we shall consider simple counting methods and use them in solving such
simple counting problems.
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Permutations And Combinations
MODULE - I
Algebra
OBJECTIVES
After studying this lesson, you will be able to :
Notes
•
find out the number of ways in which a given number of objects can be arranged;
•
state the Fundamental Principle of Counting;
•
define n! and evaluate it for defferent values of n;
•
state that permutation is an arrangement and write the meaning of n Pr ;
•
n
state that Pr =
•
show that (i ) ( n + 1) n Pn = n +1 Pn
•
state that a combination is a selection and write the meaning of n C r ;
•
distinguish between permutations and combinations;
•
n
derive Cr =
•
derive the relation n Pr = r ! nCr ;
•
verify that nCr = nCn − r and give its interpretation; and
•
derive nCr + nCn − r = n +1Cr and apply the result to solve problems.
n!
and apply this to solve problems;
(n − r )!
(ii ) n Pr +1 = ( n − r ) n Pr ;
n!
and apply the result to solve problems;
r !(n − r )!
EXPECTED BACKGROUND KNOWLEDGE
•
Number Systems
•
Four Fundamental Operations
7.1COUNTING PRINCIPLE
Let us now solve the problem mentioned in the introduction. We will writet1, t2 to denote trains
from Bangalore to Itarsi and T1, T2, T3, for the trains from Itarsi to Allahabad. Suppose I take
t1 to travel from Bangalore to Itarsi. Then from Itarsi I can takeT1 or T2 or T3. So the possibilities
are t1T1, t2T2 and t3T3 where t1T1 denotes travel from Bangalore to Itarsi by t1 and travel from
Itarsi to Allahabad by T1. Similarly, if I take t2 to travel from Bangalore to Itarsi, then the
possibilities are t2T1, t2T2 and t2T3. Thus, in all there are 6 (2 × 3) possible ways of travelling
from Bangalore to Allahabad.
Here we had a small number of trains and thus could list all possibilities. Had there been 10
trains from Bangalore to Itarsi and 15 trains from Itarsi to Allahabad, the task would have been
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Permutations And Combinations
MODULE - I
Algebra
very tedious. Here the Fundamental Principle of Counting or simply the Counting Principle
comes in use :
If any event can occur in m ways and after it happens in any one of these ways, a
second event can occur in n ways, then both the events together can occur in m× n
ways.
Notes
Example 7.1 How many multiples of 5 are there from 10 to 95 ?
Solution : As you know, multiples of 5 are integers having 0 or 5 in the digit to the extreme right
(i.e. the unit’s place).
The first digit from the right can be chosen in 2 ways.
The second digit can be any one of 1,2,3,4,5,6,7,8,9.
i.e. There are 9 choices for the second digit.
Thus, there are 2 × 9 = 18 multiples of 5 from 10 to 95.
Example 7.2 In a city, the bus route numbers consist of a natural number less than 100,
followed by one of the letters A,B,C,D,E and F. How many different bus routes are possible?
Solution : The number can be any one of the natural numbers from 1 to 99.
There are 99 choices for the number.
The letter can be chosen in 6 ways.
∴ Number of possible bus routes are 99 × 6 = 594 .
CHECK YOUR PROGRESS 7.1
1.
(a) How many 3 digit numbers are multiples of 5?
(b) A coin is tossed thrice. How many possible outcomes are there?
(c) If you have 3 shirts and 4 pairs of trousers and any shirt can be worn with any pair of
trousers, in how many ways can you wear your shirts and pairs of trousers?
(d) A tourist wants to go to another country by ship and return by air. She has a choice of
5 different ships to go by and 4 airlines to return by. In how many ways can she perform
the journey?
2.
(a) In how many ways can two vacancies be filled from among 4 men and 12 women if
one vacancy is filled by a man and the other by a woman?
(b) Flooring and painting of the walls of a room needs to be done. The flooring can be
done in 3 colours and painting of walls can be done in 12 colours. If any colour combination
is allowed, find the number of ways of flooring and painting the walls of the room.
So far, we have applied the counting principle for two events. But it can be extended to
three or more, as you can see from the following examples :
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Permutations And Combinations
MODULE - I
Algebra
Example 7.3 There are 3 questions in a question paper. If the questions have 4,3 and 2
solutionsvely, find the total number of solutions.
Solution : Here question 1 has 4 solutions,
question 2 has 3 solutions
Notes
and question 3 has 2 solutions.
∴
By the multiplication (counting) rule,
total number of solutions
= 4 × 3× 2
= 24
Example 7.4 Consider the word ROTOR. Whichever way you read it, from left to right or
from right to left, you get the same word. Such a word is known as palindrome. Find the
maximum possible number of 5-letter palindromes.
Solution : The first letter from the right can be chosen in 26 ways because there are 26
alphabets.
Having chosen this, the second letter can be chosen in 26 ways
∴ The first two letters can chosen in 26 × 26 = 676 ways
Having chosen the first two letters, the third letter can be chosen in 26 ways.
∴ All the three letters can be chosen in 676 × 26 = 17576 ways.
It implies that the maximum possible number of five letter palindromes is 17576 because the
fourth letter is the same as the second letter and the fifth letter is the same as the first letter.
Note : In Example 7.4 we found the maximum possible number of five letter
palindromes. There cannot be more than 17576. But this does not mean that there
are 17576 palindromes. Because some of the choices like CCCCC may not be
meaningful words in the English language.
Example 7.5 How many 3-digit numbers can be formed with the digits 1,4,7,8 and 9 if the
digits are not repeated.
Solution : Three digit number will have unit’s, ten’s and hundred’s place.
Out of 5 given digits any one can take the unit’s place.
This can be done in 5 ways.
... (i)
After filling the unit’s place, any of the four remaining digits can take the ten’s place.
This can be done in 4 ways.
... (ii)
After filling in ten’s place, hundred’s place can be filled from any of the three remaining digits.
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